M403L Second Exam Questions

Problem 1. Evaluate the iterated integral .

Problem 2. Consider (but do not evaluate!) the iterated integral .

a) Sketch the region of integration.

b) Rewrite the expression as an integral dx dy (That is, ``switch the order of integration''). You do NOT have to evaluate this integral, just write it down. [You may find it convenient to rewrite as .]

Problem 3. Consider the differential equation , whose general solution is .

a) Find the particular solution with y(0)=2.

b) If y(0)=2, what is the value of y(3)?

Problem 4. Consider the differential equation .

a) Find the integrating factor I(x). [Simplify your answer as much as possible]

b) Find the general solution to the differential equation. [Again, you should simplify your answer as much as possible].

Problem 5. A certain pollutant is emitted at a rate of 1,000 tons per year. It is broken down in the environment at a rate of 10% (of the amount out there) per year.

a) Write down the differential equation that describes this situation. [You do not have to solve the differential equation].

b) Is this equation of a type you (should) recognize? Describe qualitatively what happens over time.

Problem 6. a) Find the second-order Taylor polynomial that approximates the function near x=8.

b) Use this polynomial to approximate .